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If you are arranging n items in a set, the number of different permutations possible is n!.

n! is pronounced n factorial.
n! = n(n-1)(n-2)(n-3) . . . * 2 * 1

For instance,
2! = 2 * 1
3! = 3 * 2 * 1
4! = 4 * 3 * 2 * 1

Since there are 8 letters in the word vacation, the number of ways to arrange the letters is 8! or 40320. However, this problem asks for the number of different distinct ways. Since there are two letter A's in the word, the different distinct ways of arranging the two A's are indistinguishable. To find the number of distinct permutations, divide the factorial of the elements in the set by the factorial of the number of identical elements.

Thus, the number of different distinct ways to arrange the letters in the word VACATION is (40320/2) or 20,160.